3A/3B Mathematics Specialist NPR-Why can't I focus my telescope?
The following message was discussed recently on the Perth Observatory volunteers mailing list:
I tried to set up a six inch Newtonian telescope on my sister’s farm (excellent viewing!).. Unfortunately I had a problem and was in a bit of a rush.
The problem: In trying to align the sighter scope and main scope during daylight hours I could only get about 30m to 40m away from trees that I was using as a target. With this, I could not focus the main scope even though I could see the trees through it. It was possible to focus the main scope by pulling the eyepiece out about 10cm and holding it (in the air) while lining it up with the hole where it fits into the main scope – not terribly satisfactory. This surprised me, because I thought the trees were virtually at infinity and so should have been able to come to a sharp focus without pulling out the eyepiece.
Background: about Newtonian telescopes
Parallel lines converge at the focus F
A Newtonian telescope uses a parabolic mirror to produce an image of its target. The "six inch" referred to in the query is the aperture: the diameter of this mirror. A more important number (from our point of view) is the focal length of the parabolic mirror. Parallel rays of light entering the telescope will converge at a point after being reflected by the mirror. The distance of this point from the mirror is the focal length. (Astronomical targets are so far away that their light comes in rays so nearly parallel as to make no practical difference.)
The original query does not specify the focal length, but it is common for astronomical telescopes to have a focal length:aperture ration of about 8:1, so we'll work on this. For a 15cm aperture this gives us a focal length of 120cm. Referring to the diagram, this gives us VF=120cm.
Background: the optics of mirrors
Angle of reflection equals angle of incidence.
The only real physics that is needed for this problem is an understanding that when a ray is reflected, the angle of reflection is equal to the angle of incidence as shown in the diagram.
Note: 'normal' here means a line at right angles to the plane of the mirror. For a curved mirror, the normal is a line at right angles to a plane tangent to the curve at the point of reflection. Angles of incidence and reflection are usually measured relative to the normal, but for this problem you could use the angles relative to the tangent if that is more convenient.
The problem
Assuming that the focus mechanism is set to 120cm with adjustment of 4cm either way and that a distant star would be in perfect focus in the middle of this range. Should the telescope be able to focus on the trees 40m away? Do the results of the person who posted the query make sense, or should he conclude that there is something wrong with the telescope and/or eye pieces?
Show full working and give clear reasoning for your answer based on the geometry of the telescope.
Are you able to generalise this for a mirror with focal length of f aimed at a target d metres away?
Skills needed
To solve this problem you may need the following new skills, depending on your approach:
Differentiation of simple polynomials
Trigonometric identities for complementary angles, angle sum/difference and double angle
Solutions
Create a page for your solution here by putting the problem title and your name in square brackets like this:
Then save this page and click on the new link you have created to go to the new page. Edit that page and put your solution there. Sign it at the bottom with four tildes (i.e. ~). This will automatically turn into a date-stamped signature like this: - glenprideaux Jan 15, 2009
For help editing these pages, see Editing.
This page has been edited 8 times. The last modification was made by - glenprideaux on Jan 15, 2009 3:17 pm
Table of Contents
3A/3B Mathematics Specialist NPR-Why can't I focus my telescope?
The following message was discussed recently on the Perth Observatory volunteers mailing list:
I tried to set up a six inch Newtonian telescope on my sister’s farm (excellent viewing!).. Unfortunately I had a problem and was in a bit of a rush.
The problem: In trying to align the sighter scope and main scope during daylight hours I could only get about 30m to 40m away from trees that I was using as a target. With this, I could not focus the main scope even though I could see the trees through it. It was possible to focus the main scope by pulling the eyepiece out about 10cm and holding it (in the air) while lining it up with the hole where it fits into the main scope – not terribly satisfactory. This surprised me, because I thought the trees were virtually at infinity and so should have been able to come to a sharp focus without pulling out the eyepiece.
Background: about Newtonian telescopes
The original query does not specify the focal length, but it is common for astronomical telescopes to have a focal length:aperture ration of about 8:1, so we'll work on this. For a 15cm aperture this gives us a focal length of 120cm. Referring to the diagram, this gives us VF=120cm.
Background: the optics of mirrors
Note: 'normal' here means a line at right angles to the plane of the mirror. For a curved mirror, the normal is a line at right angles to a plane tangent to the curve at the point of reflection. Angles of incidence and reflection are usually measured relative to the normal, but for this problem you could use the angles relative to the tangent if that is more convenient.
The problem
Assuming that the focus mechanism is set to 120cm with adjustment of 4cm either way and that a distant star would be in perfect focus in the middle of this range. Should the telescope be able to focus on the trees 40m away? Do the results of the person who posted the query make sense, or should he conclude that there is something wrong with the telescope and/or eye pieces?
Show full working and give clear reasoning for your answer based on the geometry of the telescope.
Are you able to generalise this for a mirror with focal length of f aimed at a target d metres away?
Skills needed
To solve this problem you may need the following new skills, depending on your approach:Solutions
Create a page for your solution here by putting the problem title and your name in square brackets like this:Then save this page and click on the new link you have created to go to the new page. Edit that page and put your solution there. Sign it at the bottom with four tildes (i.e. ~). This will automatically turn into a date-stamped signature like this: -
For help editing these pages, see Editing.
This page has been edited 8 times. The last modification was made by
-